Series and Parallel Transfer Functions

The transfer function conveniently captures the algebraic structure of a filtering operation with respect to series or parallel combination:

• Transfer functions of filters in series multiply together
  If the output of filter $H_1(z)$ is given as input to filter $H_2(z)$ (series combination), as shown in Fig.6.1, the overall transfer function is $H(z)=H_1(z)H_2(z)$. Thus, the transfer functions of two filters connected in series simply multiply together.^7.1

  Figure 6.1: Series combination of transfer functions $H_1(z)$ and $H_2(z)$ to produce $H(z)=H_1(z)H_2(z)$.

  

• Transfer functions of parallel filters sum together
  If two filters $H_1(z)$ and $H_2(z)$ are driven by the same input signal, and if their outputs are summed, as depicted in Fig.6.2, this is called a parallel combination of filters $H_1(z)$ and $H_2(z)$. The transfer function of the parallel combination is then $H(z)=H_1(z)+H_2(z)$. This result follows immediately from linearity of the z transform.

  Figure 6.2: Parallel combination of transfer functions $H_1(z)$ and $H_2(z)$, yielding $H(z)=H_1(z)+H_2(z)$.